On the Existence of Positive Solutions for Some Nonlinear Boundary Value Problems II

We study a class of boundary value problems with $\phi$-Laplacian (e.g., the prescribed mean curvature equation, in which $\phi (s)=\frac{s}{\sqrt{1+s^2}}$) \begin{center} $-\left(\varphi(u’)\right)'=\lambda f(u); $\text{ on }$(-L, L),\quad u(-L)=u(L)=0,\end{center} where\lambda$and$L$ are positive parameters. For convex $f$ with $f(0)=0$, we establish various results on the exact number of positive solutions as well as global bifurcation diagrams. Some new bifurcation patterns are shown. This paper is a continuation of Pan and Xing [13], where the case $f(0)>0$ has been investigated.